In a NaCl-type crystal the anions (radius R) form an fcc lattice and touch each other along the face diagonal. Cations of radius exactly (sqrt2-1)R fill all the octahedral voids. Taking pi=3.14, the fraction of the cell volume occupied by the ions is closest to

JEE Advanced levelSingle correct optionDifficulty 3 of 3ss159

In a NaCl-type crystal the anions (radius $R$ ) form an fcc lattice and touch each other along the face diagonal. Cations of radius exactly $(2 - 1) R$ fill all the octahedral voids. Taking $π = 3.14$ , the fraction of the cell volume occupied by the ions is closest to

a = 22 R

a^{3} = 162 R^{3} = 22.63 R^{3}

. There are 4 anions and 4 cations per cell, cation radius

= 0.4142 R

. Volume of ions

= \frac{4}{3} π R^{3} [4 + 4 (0.4142)^{3}] = \frac{4}{3} π R^{3} \times 4.284

. Fraction

= 0.793